Applying the Equation of Time when calculating longitude at the Sun’s Meridian Passage.


Author’s Note:  The original issue of this post was found to contain errors which have now been corrected.  Many thanks to Jeremy Parker for bringing the errors to my attention.

Although the imaginary Mean Time gives us an accurate measurement of time, it presents the navigator with a problem.   When fixing his position by an observation of the Sun, he measures the altitude of the True Sun which keeps apparent solar time.  However, he notes the time of the observation from a deck watch that keeps mean solar time.  To enable us to connect mean solar time with apparent solar time, we have the Equation of Time which is defined as follows:

Equation of Time = Mean solar time – Apparent Solar Time

In other words, the equation of time is the difference between apparent solar time and mean solar time taken at the same instant at one place.

The equation of time can be either positive or negative depending on the time of the year.

  • The values range from approximately +15 to -15 mins. but can be as much as +64 mins.
  • The values are positive from 15th April to 14th June and from 1st September to 24th
  • The values are negative from 15th June to 31st August and from 25th December to 14th

Nautical Almanac.

extractt-eot

The Equation of Time for 00h (lower meridian) and 12h (upper meridian) for each day is printed at the foot of the Nautical Almanac daily page as shown in this extract.  The Local Mean Time of the Sun’s Meridian Passage is shown in the column to the right of the EOT. (This is the apparent time of Mer Pas adjusted for EOT to give the LMT and rounded up to the nearest minute).

If the mean time of Mer. Pas is shown to be greater than 1200 then the EOT must be negative, indicating that apparent time is slow compared to mean time.  Conversely, if the mean time of Mer. pas. is less than 1200 then EOT is positive, indicating that apparent time is ahead of mean time.

To calculate longitude we simply find the difference between LMT of Mer. Pas. and the GMT of our observation of Mer. Pas. then, by converting the time difference to arc we are able to find the difference in degrees of longitude.

Let’s try an example:

Date: 22 June.  Zone Time: 1140 (+4).  DR Pos: 320 30’N.  610 55’W.  At meridian passage, the deck watch time was 16h 08m 25.1s  and the Deck Watch Error was  -05.0s.  The daily page for that date shows that the Eqn. of Time is 02m 02s and that Mer. Pas. is 1202 indicating that EOT is negative.

Procedure:

  • Calculate time difference.

Deck Watch Time:        16h 08m 25.1s

DWE:                                                    -05.0s

GMT:                                    16h 08m 20.1s

LMT Mer.Pas:                12h  02m 00s 

Time Diff:                        04h 06m 20.1s

(Longitude West, GMT Best)

  • Convert Time to Arc

4h = 4 x 15  =            60o  00’  00”

06m = 6 ÷ 4  =             1o 30’  00”

20.1s = 20.1 ÷ 4 =      0o  05’  01″.5

Therefore, Long = 61o  35’ 01″.5 W

The navigator seldom requires the time of meridian passage to accuracies greater than one minute. Therefore, use the time listed under the “Mer. Pass.” column unless extreme accuracy is required.

Alternative Method.  Notwithstanding the slight inaccuracy caused by the rounding up of the listed time of Mer. pas., practicing navigators may prefer the above method but for students and tutors of astro navigation, it provides very little understanding of the equation of time.  To overcome this problem, it may be more beneficial for students to calculate the time of Mer. Pas themselves by applying the equation of time.  There is also the benefit of a greater degree of accuracy.

So if, in the above example, we wished to recalculate the time difference by applying the equation of time to the local apparent noon (LAN) instead of simply using the published time of Mer. Pas., we would proceed as follows:

extractt-eot

On 22 June, the EOT is 01m 55s at 00h and it is 02m 02s at 12h so the hourly rate of change is (02m 02s – 01m 55s) ÷ 12 = 0.58s.  Therefore, at 16h 00m, the EOT will be 02m 02s + (4 x 0.58s) = 02m 04.32s which we can approximate to 02m 04s.  The table shows that Mer. Pas is greater than 1200 indicating that apparent time is slow compared to mean time.  This means that  EOT is negative and must be added to the apparent time to give mean time.

  • Calculate time difference.

Deck Watch Time:        16h 08m 25.1s

DWE:                                                    -05.0s

GMT Mer. Pas:               16h 08m 20.1s

LAN:                                 12h  00m 00s 

EOT:                                          02m 04s 

LMT Mer. Pas.:              12h  02m 04s 

Time Diff (GMT – LMT) = 04h 06m 16.1s

(Longitude West, GMT Best) 

  • Convert Time to Arc

4h = 4 x 15  =       60o  00’ 00”

06m = 06 ÷ 4  =       1o 30’  00”

16.1s = 16.1 ÷ 4 =  0o  04’  01″.5

Therefore, Long = 61o  34’ 01″.5 W

Links:  Understanding Meridian Passage,   Meridian Passage Short Method,  Meridian Passage Long Method    What is the point of meridian passage?     Zone Time,    Local Hour Angle and Greenwich Hour Angle,   Converting GMT to GHA ,  Altitude Corrections

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Astro Navigation Demystified at Amazon.com

Astro Navigation Demystified at Amazon.uk

Applying Mathematics to Astro Navigation at Amazon .com

Applying Mathematics to Astro Navigation at Amazon .uk

Astronomy for Astro Navigation at Amazon.com

Astronomy for Astro Navigation at Amazon.uk

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